Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
466 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#884 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#413 | C2 |
| socle | C2×C3 | C2×#121 |
| hypercenter | #413 | D3 |
| hypocenter | C3 | #3068 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #413 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5822 | C2 | semidirect |
| 2. C2×#949 | C2 | semidirect |
| 3. C2×#908 | C2 | semidirect |
| 4. #5745 | C2 | semidirect |
| 5. C3×#413 | C2 | semidirect |
| 6. C2×#823 | C2 | semidirect |
| 7. #5124 | C2 | semidirect |
| 8. #4754 | C2 | semidirect |
| 9. #4749 | C2 | semidirect |
| 21. #942 | C22 | semidirect |
| 22. C2×#215 | C22 | semidirect |
| 23. #932 | C22 | semidirect |
| 24. C2×#213 | C22 | semidirect |
| 29. #907 | C22 | semidirect |
| 34. C3×#101 | C22 | semidirect |
| 35. C3×#100 | C22 | semidirect |
| 36. #823 | C22 | semidirect |
| 37. #822 | C22 | semidirect |
| 45. #413 | D3 | semidirect |
| 117. #101 | C2×D3 | semidirect |
| 119. #100 | C2×D3 | semidirect |
| 287. C3 | #3068 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5822 | 2 |
| C2×#949 | 2 |
| C2×#908 | 2 |
| #5745 | 2 |
| C3×#413 | 2 |
| C2×#823 | 2 |
| #5124 | 2 |
| #4754 | 2 |
| #4749 | 2 |
| #3068 | 3 |